Geometry optimization
Fixed cell
We use DFTK and the GeometryOptimization package to find the minimal-energy bond length of the $H_2$ molecule. First we set up an appropriate DFTKCalculator (see AtomsCalculators integration), for which we use the LDA model just like in the Tutorial for silicon in combination with a pseudodojo pseudopotential (see Pseudopotentials).
using DFTK
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf")
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials), # model_DFT keyword arguments
basis_kwargs = (; kgrid=[1, 1, 1], Ecut=10) # PlaneWaveBasis keyword arguments
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[1, 1, 1])Next we set up an initial hydrogen molecule within a box of vacuum. We use the parameters of the equivalent tutorial from ABINIT and DFTK's AtomsBase integration to setup the hydrogen molecule. We employ a simulation box of 10 bohr times 10 bohr times 10 bohr and a pseudodojo pseudopotential.
using LinearAlgebra
using Unitful
using UnitfulAtomic
r0 = 1.4 # Initial bond length in Bohr
a = 10.0 # Box size in Bohr
cell_vectors = [[a, 0, 0]u"bohr", [0, a, 0]u"bohr", [0, 0, a]u"bohr"]
h2_crude = periodic_system([:H => [0, 0, 0.]u"bohr",
:H => [r0, 0, 0]u"bohr"],
cell_vectors)FlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [ 0, 0, 0]u"a₀")
Atom(H, [ 1.4, 0, 0]u"a₀")
Finally we call minimize_energy! to start the geometry optimisation. We use verbosity=2 to get some insight into the minimisation. With verbosity=1 only a summarising table would be printed and with verbosity=0 (default) the minimisation would be quiet.
using GeometryOptimization
results = minimize_energy!(h2_crude, calc; tol_forces=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110115723205 -0.82 0.80 7.0 5.89s
2 -1.117173382728 -2.15 -1.82 0.80 1.0 2.37s
3 -1.117249600940 -4.12 -2.67 0.80 1.0 19.7ms
4 -1.117251093861 -5.83 -3.38 0.80 1.0 19.4ms
5 -1.117251184659 -7.04 -4.03 0.80 1.0 19.5ms
6 -1.117251190129 -8.26 -5.02 0.80 2.0 21.9ms
7 -1.117251190138 -11.06 -5.37 0.80 1.0 48.3ms
8 -1.117251190143 -11.32 -6.25 0.80 1.0 20.7ms
9 -1.117251190143 -12.60 -6.56 0.80 2.0 23.0ms
10 -1.117251190143 -13.91 -6.89 0.80 1.0 29.2ms
11 -1.117251190143 -14.12 -8.04 0.80 1.0 21.2ms
12 -1.117251190143 -15.35 -8.07 0.80 2.0 23.0ms
13 -1.117251190143 + -15.05 -8.58 0.80 1.0 24.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -8.90 0.80 1.0 4.17s
2 -1.117251190143 -14.51 -9.83 0.80 1.0 86.6ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 23.8s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997926 -2.46 0.80 1.0 57.7ms
2 -1.117520586051 -5.59 -3.49 0.80 1.0 18.6ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 18.6ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 18.9ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 34.8ms
6 -1.117520613465 -12.13 -6.42 0.80 1.0 19.7ms
7 -1.117520613465 -13.36 -7.39 0.80 1.0 19.9ms
8 -1.117520613465 -14.88 -8.19 0.80 1.0 23.3ms
9 -1.117520613465 -14.65 -8.96 0.80 1.0 20.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264592 -1.69 0.80 1.0 13.0ms
2 -1.118339043931 -4.04 -2.70 0.80 1.0 18.5ms
3 -1.118340011110 -6.01 -3.44 0.80 1.0 24.1ms
4 -1.118340028551 -7.76 -4.36 0.80 1.0 19.0ms
5 -1.118340029136 -9.23 -5.12 0.80 1.0 21.8ms
6 -1.118340029162 -10.57 -5.62 0.80 1.0 19.7ms
7 -1.118340029164 -11.73 -6.68 0.80 1.0 22.4ms
8 -1.118340029164 -13.54 -7.58 0.80 1.0 20.7ms
9 -1.118340029164 + -14.88 -8.12 0.80 1.0 22.9ms
10 -1.118340029164 -15.65 -8.55 0.80 1.0 20.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029164 -9.05 0.80 1.0 13.1ms
2 -1.118340029164 -14.65 -10.36 0.80 1.0 18.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029164 -11.04 0.80 1.0 12.9ms
2 -1.118340029164 -14.75 -11.41 0.80 1.0 25.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029164 │ │ 0.00297055 │ 2.90s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 20.0ms
2 -1.118346864757 -6.84 -4.11 0.80 1.0 18.5ms
3 -1.118346866234 -8.83 -4.86 0.80 1.0 18.6ms
4 -1.118346866260 -10.59 -5.78 0.80 1.0 89.8ms
5 -1.118346866261 -12.08 -6.55 0.80 1.0 19.5ms
6 -1.118346866261 -13.44 -7.04 0.80 1.0 568ms
7 -1.118346866261 -14.65 -8.13 0.80 1.0 22.0ms
8 -1.118346866261 + -14.75 -8.95 0.80 1.0 20.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 12.8ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.2ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 18.5ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 18.6ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.2ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 19.3ms
7 -1.118356101386 -13.66 -7.65 0.80 1.0 20.1ms
8 -1.118356101386 -14.75 -8.45 0.80 1.0 19.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 12.9ms
2 -1.118356101386 -15.35 -9.82 0.80 1.0 19.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 17.4ms
2 -1.118356101386 -14.57 -11.46 0.80 1.0 25.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 1.14s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.0ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.4ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 18.6ms
4 -1.118356102877 -14.12 -7.62 0.80 1.0 18.9ms
5 -1.118356102877 + -14.51 -8.39 0.80 1.0 20.3ms
6 -1.118356102877 + -14.88 -8.89 0.80 1.0 19.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 21.2ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.5ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.5ms
4 -1.118356105120 -13.19 -7.09 0.80 1.0 18.9ms
5 -1.118356105120 + -14.65 -7.86 0.80 1.0 19.2ms
6 -1.118356105120 -14.61 -8.35 0.80 1.0 19.4ms
7 -1.118356105120 + -14.61 -9.45 0.80 1.0 19.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 13.0ms
2 -1.118356105120 -14.37 -11.02 0.80 1.0 18.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 12.7ms
2 -1.118356105120 -14.09 -12.44 0.80 1.0 22.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.10624e-8 │ 455ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
Structure after optimisation (note that the atom has wrapped around)
results.systemFlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [-0.0431829, -9.21391e-35, 0]u"a₀")
Atom(H, [ 1.44318, -5.3008e-35, 0]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.48636586447076 a₀
Our result (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.
Variable cell
Recent versions of GeometryOptimization support cell optimization as well by passing variablecell=true to minimize_energy!.
For a plane-wave code like DFTK, variable cells pose an additional challenge: changes to the cell's size affect the used plane waves, leading to discontinuities in the energy and stresses. This can cause the geometry optimization to struggle and/or fail to converge.
A practical strategy to overcome this problem is Energy cutoff smearing. As a demonstration let us find the optimal lattice constant of silicon.
Like before we define a calculator, this time with a kinetic_blowup set to use energy cutoff smearing:
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials,
kinetic_blowup=BlowupCHV()),
basis_kwargs = (; kgrid=[2, 2, 2], Ecut=10)
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[2, 2, 2])And here is our starting silicon structure:
a = 10.0u"bohr" # Approximate Silicon lattice constant
cell_vectors = a/2 * [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
initial_silicon = periodic_system([:Si => ones(3)/8,
:Si => -ones(3)/8],
cell_vectors;
fractional=true)FlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [ 0 5 5;
5 0 5;
5 5 0]u"a₀"
Atom(Si, [ 1.25, 1.25, 1.25]u"a₀")
Atom(Si, [ -1.25, -1.25, -1.25]u"a₀")
We now minimize, passing variablecell=true:
using GeometryOptimization
results = minimize_energy!(initial_silicon, calc; variablecell=true,
tol_virial=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.335211418154 -0.80 0.80 7.7 758ms
2 -8.340937879590 -2.24 -1.69 0.80 1.3 617ms
3 -8.341208199402 -3.57 -2.78 0.80 2.3 32.1ms
4 -8.341214222799 -5.22 -3.30 0.80 3.3 68.7ms
5 -8.341214267266 -7.35 -4.00 0.80 1.7 28.7ms
6 -8.341214272488 -8.28 -5.76 0.80 2.7 34.5ms
7 -8.341214272518 -10.53 -6.08 0.80 4.0 42.4ms
8 -8.341214272518 -12.70 -6.84 0.80 1.7 28.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.31 0.80 1.0 632ms
2 -8.341214272518 + -14.27 -8.32 0.80 1.0 25.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.89 0.80 1.0 22.0ms
2 -8.341214272518 -14.75 -9.13 0.80 1.0 24.9ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350386794649 -1.20 0.80 9.0 69.7ms
2 -8.351864885793 -2.83 -1.88 0.80 1.3 28.2ms
3 -8.351926043272 -4.21 -2.77 0.80 3.3 38.0ms
4 -8.351928510411 -5.61 -3.66 0.80 3.3 39.0ms
5 -8.351928514534 -8.38 -4.98 0.80 2.7 35.5ms
6 -8.351928514653 -9.92 -6.00 0.80 4.3 45.7ms
7 -8.351928514653 -12.71 -6.90 0.80 4.0 41.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353382374419 -1.52 0.80 7.7 61.5ms
2 -8.353724620849 -3.47 -2.21 0.80 1.0 25.4ms
3 -8.353757365929 -4.48 -2.98 0.80 4.0 40.3ms
4 -8.353757523281 -6.80 -4.20 0.80 2.3 32.3ms
5 -8.353757531691 -8.08 -5.01 0.80 3.3 40.9ms
6 -8.353757531719 -10.56 -5.82 0.80 2.7 32.5ms
7 -8.353757531720 -11.85 -7.19 0.80 3.3 38.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531720 -8.41 0.80 1.0 22.1ms
2 -8.353757531720 -14.15 -9.31 0.80 1.0 24.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531720 -9.57 0.80 1.0 22.6ms
2 -8.353757531720 -14.75 -9.96 0.80 1.0 25.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531720 -10.11 0.80 1.0 23.8ms
2 -8.353757531720 + -Inf -10.29 0.80 1.0 25.4ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531720 │ │ 2.50603e-32 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353694734595 -3.18 0.80 8.0 64.8ms
2 -8.353756549355 -4.21 -3.71 0.80 1.0 25.2ms
3 -8.353770531524 -4.85 -3.66 0.80 6.0 54.0ms
4 -8.353770536740 -8.28 -3.95 0.80 1.0 25.5ms
5 -8.353770536551 + -9.72 -4.18 0.80 1.7 28.2ms
6 -8.353770537312 -9.12 -4.70 0.80 1.0 25.7ms
7 -8.353770537385 -10.14 -5.09 0.80 2.0 34.3ms
8 -8.353770537397 -10.91 -6.96 0.80 1.7 29.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353778148714 -2.00 0.80 8.3 62.6ms
2 -8.353878342153 -4.00 -2.70 0.80 1.0 29.8ms
3 -8.353895049757 -4.78 -3.32 0.80 4.0 42.2ms
4 -8.353895106890 -7.24 -4.30 0.80 2.0 31.0ms
5 -8.353895110715 -8.42 -5.13 0.80 3.3 42.4ms
6 -8.353895110753 -10.42 -5.91 0.80 3.0 36.0ms
7 -8.353895110753 -12.04 -6.86 0.80 2.7 36.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110753 -7.51 0.80 1.0 26.6ms
2 -8.353895110753 -14.75 -8.59 0.80 1.0 25.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110753 -9.22 0.80 1.0 22.4ms
2 -8.353895110753 + -Inf -9.38 0.80 1.0 29.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110753 -9.67 0.80 1.0 22.3ms
2 -8.353895110753 -14.75 -9.99 0.80 1.0 29.7ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110753 │ -3.86 │ 4.31586e-32 │ 0.000646202 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353751009272 -3.62 0.80 8.3 67.0ms
2 -8.353834190318 -4.08 -3.78 0.80 1.0 25.6ms
3 -8.353895211567 -4.21 -4.04 0.80 6.7 57.2ms
4 -8.353895212007 -9.36 -4.26 0.80 1.3 26.9ms
5 -8.353895212465 -9.34 -4.82 0.80 1.3 30.0ms
6 -8.353895212498 -10.49 -4.99 0.80 1.7 29.8ms
7 -8.353895212504 -11.17 -5.44 0.80 1.0 26.1ms
8 -8.353895212506 -11.76 -6.22 0.80 2.3 32.0ms
9 -8.353895212506 -12.71 -6.25 0.80 3.0 40.4ms
10 -8.353895212506 -14.27 -6.25 0.80 1.0 26.8ms
11 -8.353895212506 -13.85 -6.62 0.80 1.0 26.9ms
12 -8.353895212506 -13.97 -6.82 0.80 1.0 31.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353800649500 -3.36 0.80 7.7 60.3ms
2 -8.353874222178 -4.13 -3.76 0.80 1.0 30.7ms
3 -8.353895362272 -4.67 -4.08 0.80 6.0 53.4ms
4 -8.353895362606 -9.48 -4.54 0.80 1.0 25.7ms
5 -8.353895362676 -10.16 -5.00 0.80 1.7 28.4ms
6 -8.353895362685 -11.03 -5.68 0.80 2.0 33.9ms
7 -8.353895362686 -12.10 -6.99 0.80 3.0 35.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.92 0.80 1.0 22.0ms
2 -8.353895362686 -14.15 -8.25 0.80 1.0 29.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.60 0.80 1.0 22.8ms
2 -8.353895362686 + -Inf -8.95 0.80 1.0 25.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.16 0.80 1.0 22.6ms
2 -8.353895362686 -14.75 -9.52 0.80 1.0 25.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ │ 1.71719e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [5.08749e-36 5.27783 5.27783;
5.27783 5.08749e-36 5.27783;
5.27783 5.27783 5.08749e-36]u"a₀"
Atom(Si, [ 1.31946, 1.31946, 1.31946]u"a₀")
Atom(Si, [-1.31946, -1.31946, -1.31946]u"a₀")
Since here the cell was rescaled but its shape did not change, we directly extract the optimal lattice constant from one of the cell vectors:
using AtomsBase
amin = AtomsBase.cell_vectors(results.system)[1][2]*2
println("Optimal lattice constant: ", amin)Optimal lattice constant: 10.555662922537172 a₀
Note that while for silicon the positions of the atoms are fixed by symmetry, in general a variable cell optimization will try to optimize both the cell and the positions of the individual atoms.